Annual $1000.00 raise vs semi-annual $300.00 raise

Cecil's Columns/Staff Reports
1

Sorry folks, I must’ve somehow hit the ‘Submit’ button before I was finished. Could a mod please delete my OP and replace it with this post? Thanks.

Anyway, from the top:

Blasphemous as it may be, I think Cecil has this one wrong. First off, check out this column. All done? Cool.

Now, what I don’t understand is that a semi-annual raise of $300.00 would be the same as an annual raise of $600.00 dollars, right? Cecil says as much, here:

So far we agree. A semi-annual $300.00 raise is the same as an annual $600.00 raise. But in the very same sentence Cecil says:

Here’s where I get confused. We’ve established that the total annual wage hike for a person taking a semi-annual $300.00 raise is $600.00 dollars. Two raises per year, 2x300 = 600. Why, then, does Cecil argue that the person taking the semi-annual $300.00 raise would get two raises of $600.00 dollars. We’ve already established that $600.00 is the annual raise. If he has two of them per year, it’s no longer annual.

Bottom line, the wages over the next few years would map out like this:



                              A                       B

01/01/2006  - $10,000.00       $10,000.00

01/01/2007  - $10,600.00       $11,000.00

01/01/2008 - $11,200.00        $12,000.00

01/01/2009 - $11,800.00        $13,000.00

01/01/2010 - $12,400.00        $14,000.00

01/01/2011 - $13,000.00        $15,000.00
-------------------------------------------------------------------

6 YEAR TOTAL: $69,000.00      $75000.00

At no point does person A get any advantage over person B, within the parameters of the question. Where am I (or maybe even Cecil :slight_smile: ) going wrong?

2

It’s very confusing, (and I still maintain that nobody would naturally interpret ‘semi-annual $300 raise’ this way,) but the way Cecil and Marilyn interpret it, it does come out to $1200 raise per year, more or less.

  • One doubling factor is the fact that the raise is in the semi-annual pay. If you get your semiannual pay increased by $300 once per year, that comes out to an increase in $600 in your annual pay, because your annual pay is twice your semiannual pay.
  • The other doubling factor is that you get the raise twice per year.

Cecil says that “In the first six months you’d make $5,000, the second six months $5,300, the third six months $5,600, and so on.” That makes the correct annual pay figures for the A column:

row 1 - 10,300. (5000 + 5300)
row 2 - 11,500. (5600 + 5900)
row 3 - 12,700. (6200 + 6500)
row 4 - 13,900. (6800 + 7100)
row 5 - 15,100. (7400 + 7700)
row 6 - 16,300. (8000 + 8300)

Does this make things clearer??

3

Took me a full day to figure out WHY Marilyn’s answer was wrong. That it’s wrong is obvious, but I couldn’t quite put my finger on it. What I finally settled on is similar to chrisk’s comment. The trick is that Marilyn incremented the wrong number. If you increment the annual salary by $300, then divide that into periodic payments (monthly, biweekly, etc.), you get the expected result. That is, the semi-annual raise takes an early lead because it kicks in earlier, but the lines cross fairly quickly and, by the end of year two, the annual raise takes the lead and pulls away.

Stated a little differently, using slightly different numbers (same orders of magnitude, but easier for doing the calculations). Suppose the base is $12,000 per year; Plan A is a $720 annual raise commencing on the first anniversary; Plan B is a $240 semi-annual starting after six months. Marilyn incremented (in effect) the $6,000 semi-annual salary. Why stop there? Why not increment the $1,000 monthly salary? Now, with the first raise, you’re making $1300 a month, $1600 a month after a year, etc. Obviously, beats the daylights out of the annual raise. Assuming the boss is stupid enough to compute it this way.

4

Incidentally, in my company, raises are rarely spoken of in dollar value terms at all… but percentages. Which nicely steps around the issue of if a given raise is per annum, per quarter, per paycheque, etcetera. It all comes out to the same thing.

:slight_smile:

5

Much. Thanks guys.

6

Oh, and thanks bibliophage for sorting out my OP.

7

Oops! Wrote that when nearly late to work and made a boo-boo. :smack: I realize this thread is probably over, but just in case not, I’ll correct the error. After positing a similar scenario with slightly different numbers, I mixed apples and oranges, i.e., numbers from both formulations. So, the third-to-last sentence should have read, “Now, with the first raise, you’re making $1240 a month, $1480 a month after a year, etc.”

Having gone this far, I’ll go back and explicate the “lines cross” comment (for which I didn’t have enough time this morning). Using the reformulated scenario ($12,000 base, $240 semi-annual raise v. $720 annual), after six months, Plan B goes up to a monthly salary of $1020. For six months, this is better than Plan A. At the first anniversary, Plan A goes to a monthly salary of $1060, while Plan B goes to $1040. Over the next six months, Plan A catches up with Plan B, i.e., both will have paid the same amount, although Plan B is better because it ponied up the money sooner. At this point, Plan B goes to a monthly salary of $1060, so the two are equivalent. (On reflection, I should have said that the lines merge at eighteen months, rather than cross at two years). On the second anniversary, Plan A goes to $1120 monthly, while Plan B goes to $1080; after another six months, Plan B goes to $1100. Thus, Plan A is now ahead, and its lead widens with each passing year. Which is exactly what we would expect.

IOW, I think Cecil was kind to Marilyn. In a problem of this type, if you reach a counterintuitive conclusion, you should go back and figure out where you went sideways. Cecil spotted the error and identified it in his column. Marilyn did not. Or perhaps she did, but was so pleased with the counterintuitive conclusion that she chose not to mention why it was fallacious. That’s one reason why I prefer Cecil to Marilyn. He gives us the Straight Dope.

8

I took it to mean that a single $300 raise applied to your semiannual income - that is, your $300 raise meant $300 more, twice a year for a total annual raise of $600. That’s with only one $300 raise to your semiannual pay. Adding another one brings it to an effective $1200 annual raise. In other words, what chrisk said.

If you give her the benefit of this interpretation (which was what Cecil suggested), then Marilyn’s answer is correct.

9

06 5000 5000
12 5300 5000
18 5600 5600
24 5900 5600
30 6200 6200
36 6500 6200
42 6800 6800
48 7100 6800
54 7400 7400
60 7700 7400
66 8000 8000
72 8300 8000

$79800 $78000

10

I interpreted it their way right off the bat.

-FrL-

11

wuh?
shouldn’t it be
06 5000 0
12 5300 11000
18 5600 0
24 5900 12000
30 6200 0
36 6500 13000
42 6800 0
48 7100 14000
54 7400 0
60 7700 15000
66 8000 0
72 8300 16000
$79800 $81000
as the left column is the ‘increase 300 every six months’ and the right is the ‘increase 1000 every 12 months’… or am i misinterpretting yet again?
78 8600 0
84 8900 17000
90 9200 0
96 9500 18000
116000 116000
meaning you have to wait 8 years before you actually see a benefit? what happened to being $700 ahead after three years?!? at the end of the third year, col.A makes 12700, while col.B makes 13000. you have to wait until the end of the fifth year before col.A makes more per year than col.B, and that’s just when they start making up for all the money they’d missed out on up to that point!.. it’s still a good idea, but in this age of switching jobs every three years for “something better”, what are the odds you’re still going to be working for this same boss-man in 8 years, when your decision has finally started paying off?

is my math wrong?.. and how?

~buttfedge

12

ahhh… now i’m just wasting space…
i notice the first 6 months in col.A, you’re still making the 5000, meaning the first twelve months in col.B, you can only make 10000, to be fair. not 11000, like i put… my bad…
taking that extra thousand away, at the end of the third year, col.A DOES make $700 more than col.B, however, it’s misleading to think “now i’m making more, i’m ahead!”, when you have to take the lost money in the first three years… but that’s really just nit-picking, so i’ll leave it alone… sorry…
~buttfedge

13

Umm, I’m pretty sure someguyintheusa’s chart, using the formulation of the problem from the original column, should say this:

06 5000 5000
12 5300 5000
18 5600 5500
24 5900 5500
30 6200 6000
36 6500 6000
42 6800 6500
48 7100 6500
54 7400 7000
60 7700 7000
66 8000 7500
72 8300 7500
IOW, using Marilyn’s interpretation, Plan B (first column) is defintely better than Plan A, and continues to get better however long you extend the series.